First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper addresses the problem of robustly estimating the low-dimensional subspace of contaminated observations when the observations are inherently coherent. Performance goes worse with increasing data coherence is a standard theoretical bottleneck of previous RPCA methods. This paper, however, circumvents this problem in a clever manner. Considering that such cluster structure is rather common in realistic data, solving this issue is certainly significantly meaningful.

The recently established RPCA [4] method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades.

The recently established RPCA [4] method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades.

In sequential hypothesis testing, Generalized Binary Search (GBS) greedily chooses the test with the highest information gain at each step. It is known that GBS obtains the gold standard query cost of $O(\log n)$ for problems satisfying the $k$-neighborly condition, which requires any two tests to be connected by a sequence of tests where neighboring tests disagree on at most $k$ hypotheses. In this paper, we introduce a weaker condition, split-neighborly, which requires that for the set of hypotheses two neighbors disagree on, any subset is splittable by some test. For four problems that are not $k$-neighborly for any constant $k$, we prove that they are split-neighborly, which allows us to obtain the optimal $O(\log n)$ worst-case query cost.

The recently established RPCA method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades. We show that the challenges raised by coherent data (i.e., data with high coherence) could be alleviated by Low-Rank Representation (LRR)~\cite{tpami_2013_lrr}, provided that the dictionary in LRR is configured appropriately. More precisely, we mathematically prove that if the dictionary itself is low-rank then LRR is immune to the coherence parameter which increases with the underlying cluster number. This provides an elementary principle for dealing with coherent data and naturally leads to a practical algorithm for obtaining proper dictionaries in unsupervised environments. Experiments on randomly generated matrices and real motion sequences verify our claims. See the full paper at arXiv:1404.4032.